Prove rigorously that an event E, independent of all other events, has probability equal to either 0 or 1.

Answer 1

If you flip a fair coin, the probability of heads is 1/2 or 0.5

But once you have flipped it, the odds now are either 1 or 0 that it's heads. Either it is heads or it isn't. P v ~P is a logical truism, a tautology. H v ~H. If you're into hard determinism, everything is fully determined, since he Big Bang, and the coin was fully determined to be either heads or not heads before you flipped it.

Still, even if you believe in a random universe, with quantum physics randomly producing events, once the event has happened, there is no longer any indeterminism, the coin is either heads or not heads. However, if you like Schrodinger's cat, then the cat is either dead or alive until you observe it. The observer is needed to have the wave function collapse, one way or the other. Maybe that applies to coins. It may be neither heads or tails until someone observes it.

Frequentist statistics would assign P(H)=1/2. Bayesian statistics would use Bayes' Theorem P(H/F)P(F)=P(F/H)P(H) which is a tautology, always true. P(H) is the a priori probability of heads or 1/2.

P(H/F) is the probability of heads after the flip. P(F/H) = 1. P(F/H)P(H) = 1 x 1/2 = 1

P(H/F) is the posterior probability of heads, once the coin is flipped. IF the a priori probability of flipping a coin is 1/2, which is the standard presupposition if there are 2 alternatives, flipping or not flipping, then the P(H/F) = 1.